The definition of distributed testing for \(H\) -freeness requires that if there is no instance of \(H\)  in the graph then all the nodes output true, but instead of requiring at least one node to output false in case there is an instance of \(H\)  as would be an analog to the detection problem, testing only requires at least one node to output false in case the graph is far from being \(H\) -free. Here, being \(\epsilon \) -far from having a property is the same as its standard definition by Goldreich, Goldwasser, and Ron [1]}, and means that no matter which \(\epsilon \) -fraction of the graph is changed (removed, in the case of \(H\) -freeness), the resulting graph must still have a copy of \(H\) . For triangles and other small subgraphs, this typically means that there are many instances of \(H\)  in a graph that is far from being \(H\) -free.
